Utility-based valuation for underwater employee stock options

In this report, we explore the theory behind utility-based valuation of stock options. In particular, we focus on the underwater employee stock options, which give rise to an incomplete-market setting. We begin with basic concepts and terminology in stock-option pricing. Then, we review the valuation by replication process both in the binomial model and the Black-Scholes model. These two methods apply to valuation in the complete-market setting. Then we introduce the concept of utility function and utility maximization in the context of portfolio allocation. An example is worked out to demonstrate how to solve the optimization problem subject to a portfolio constraint. In the end, we explore indifference pricing, i.e., utility-based valuation of stock options in an incomplete single-period binomial model.